Differential Equations Lecture Work Solutions 110

Differential - 1 r 2 utt c2(urr ur ur(a t = 0 u(r 0 =(r ut(r 0 = 0 1 T R c 2(R R T = 0 r R 1R T r = = c2 T R T c2 T = 0 R 1 R R = 0 r 1(r R R = 0 r

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2. u tt c 2 ( u rr + 1 r u r ) u r ( a, t )=0 u ( r, 0) = α ( r ) u t ( r, 0) = 0 ¨ TR c 2 ( R 0 + 1 r R 0 ) T =0 ¨ T c 2 T = R 0 + 1 r R 0 R = λ ¨ T + λc 2 T =0 R 0 + 1 r R 0 | {z } + λR =0 1 r ( rR 0 ) 0 + λR =0 multiply by r 2 r ( rR 0 ) 0 + λr 2 R =0 | R (0) | < R 0 ( a )=0 This is Bessel’s equation with µ =0 R n ( r )= J 0 ( ± λ n r ) where λ n J 0 0 ( λ n a )=0 gives the eigenvalues λ n u ( r, t )= X n =1 ² a n cos ± λ n ct + b n sin c ± λ n t ³ J 0 ( ± λ n r ) α ( r )= X n =1 a n J 0 ( ± λ n
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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